Problems

In Problems 1-16, find the increment and total differential.

In Problems 17-22, express Δz in the form Δz = dz + ε₁ Δx + ε₂ Δy.

In Problems 23-40 find the tangent plane at the given point.

41            Show that if z is a linear function of x and y, z = ax + by + c, then Δz = dz at every point (x, y).

42            Let f(x,y) =

Show that at (0,0)

(a)    f(x, y) is not continuous;

(b)    f_x(0, 0) and f_y(0,0) exist;

(c)    f(x, y) is not smooth.

43            Let f(x, y) = . Prove that at the point (0,0),

(a)     f (x, y) is continuous;

(b)    f_x(0,0) and f_y(0,0) exist;

(c)    f(x, y) is not smooth;

(d)    Δz is not infinitely close to dz compared to

44            Let f(x, y) = |xy|. Show that at (0,0),

(a)    f(x, y) is continuous;

(b)   f_x(0,0) and f(0, 0) exist;

(c)    f(x, y) is not smooth;

(d)    Δz is infinitely close to dz compared to Δs.

45            Let f(x, y) = |x| + |y|. Show that at (0, 0),

(a)    f(x, y) is continuous;

(b)    f_x(0, 0) and f_y(0,0) do not exist.